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$ΔI=1/2$ rule for kaon decays derived from QCD infrared fixed point

This article gives details of our proposal to replace ordinary chiral $SU(3)_L\times SU(3)_R$ perturbation theory $χ$PT$_3$ by 3-flavor chiral-scale perturbation theory $χ$PT$_σ$. In $χ$PT$_σ$, amplitudes are expanded at low energies and small $u,d,s$ quark masses about an infrared fixed point $α^{}_\mathrm{IR}$ of 3-flavor QCD. At $α^{}_\mathrm{IR}$, the quark condensate $\langle \bar{q}q\rangle_{\mathrm{vac}} \not= 0$ induces nine Nambu-Goldstone bosons: $π, K, η$ and a $0^{++}$ QCD dilaton $σ$. Physically, $σ$ appears as the $f_{0}(500)$ resonance, a pole at a complex mass with real part $\lesssim m_K$. The $ΔI=1/2$ rule for nonleptonic $K$-decays is then a consequence of $χ$PT$_σ$, with a $K_Sσ$ coupling fixed by data for $γγ\rightarrowππ$ and $K_{S} \to γγ$. We estimate $R_\mathrm{IR} \approx 5$ for the nonperturbative Drell-Yan ratio $R = σ(e^{+}e^{-}\rightarrow\mathrm{hadrons})/ σ(e^{+}e^{-}\rightarrowμ^{+}μ^{-})$ at $α^{}_\mathrm{IR}$, and show that, in the many-color limit, $σ/f_0$ becomes a narrow $q\bar{q}$ state with planar-gluon corrections. Rules for the order of terms in $χ$PT$_σ$ loop expansions are derived in Appendix A, and extended in Appendix B to include inverse-power Li-Pagels singularities due to external operators. This relates to an observation that, for $γγ$ channels, partial conservation of the dilatation current is not equivalent to $σ$-pole dominance.